Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels

Abstract

Tensor-product approximation provides a convenient tool for efficient numerical treatment of high-dimensional problems that arise, in particular, in electronic structure calculations in Rd. In this work we apply tensor approximation to the Galerkin representation of the Newton and Yukawa potentials for a set of tensor-product, piecewise polynomial basis functions. To construct tensor-structured representations, we make use of the well-known Gaussian transform of the potentials, and then approximate the resulting univariate integral in R by special sinc-quadratures. The novelty of the approach lies on the optimisation of the quadrature parameters that allows to reduce dramatically the initial tensor-rank obtained by the standard sinc-quadratures. The numerical experiments show that this approach gives tensor-ranks close to the optimal in 3D computations on large spatial grids and with linear complexity in the univariate grid size. Particularly, this scheme becomes attractive for the multiple calculation of the Yukawa potential when the exponents in Gaussian functions vary during the computational process.